Hadamard gap series in weighted-type spaces on the unit ball

Abstract

We give a sufficient and necessary condition for an analytic function $f\left(z\right)$ on the unit ball $\mathbb{B}$ in ${\mathbb{C}}^n$ with Hadamard gaps, that is, for $f\left(z\right)={\sum }_{k=1}^{\infty }{P}_{n_k}\left(z\right)$ where $P_{n_k}\left(z\right)$ is a homogeneous polynomial of degree $n_k$ and $n_{k+1}/n_k\ge c>1$ for all $k\in \mathbb{N}$, to belong to the weighted-type space $H_{\mu }^{\infty }$ and the corresponding little weighted-type space $H_{\mu ,0}^{\infty }$ under some condition posed on the weighted funtion $\mu$. We also study the growth rate of those functions in $H_{\mu }^{\infty }$.